Infinite series

As 冷凝夜 say, this is related to exponential integral, as follow:
First note that e^x = 1+ x/1! + x^2 / 2! + x^3/3! + ...
and your expression looks like something after integraion, we have some terms like x^n / n
So differentiate your expression, we get:
1/1! + x/2! + x^2/3! + x^3/4! + ...
= (e^x - 1) / x
So that your original expression is ∫ (e^x - 1) / x dx
This exponential integral∫ (e^x - 1) / x dx has no closed form, so this is the final answer.

if that's x/(1!) + x^2/[(2!)] + x^3/(3!) + x^4/(4!) + ......... instead, it's simply an expansion of e^x under Taylor. But this one, including Euler constant and Incomplete Gamma Function, I can't really help you. Maybe you can have a look at here:
http://mathworld.wolfram.com/IncompleteGammaFunction.html

This is related to the exponential integral, which cannot be expressed by using finitely many elementary functions.